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Expected Value Calculator

Compute the expected value of a discrete probability distribution from its outcomes and probabilities — with the variance and standard deviation.

Outcomes / values

Probabilities

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How it works

E(X) = Σ x · p(x) · Variance = Σ p(x) · (x − E(X))² · σ = √Variance

The expected value is the probability-weighted average of every possible outcome — the number you would converge on if the experiment were repeated endlessly. Each outcome is multiplied by its probability and the products are summed, so likelier outcomes pull the average toward themselves. The variance then measures how far outcomes typically stray from that expected value, again weighted by probability, and its square root is the standard deviation. The probabilities must form a valid distribution that sums to 1.

Worked example

For outcomes 10, 20, 30, 40 with probabilities 0.1, 0.2, 0.3, 0.4, the expected value is 10·0.1 + 20·0.2 + 30·0.3 + 40·0.4 = 1 + 4 + 9 + 16 = 30. The variance is 0.1·(10−30)² + 0.2·(20−30)² + 0.3·0 + 0.4·(40−30)² = 40 + 20 + 0 + 40 = 100, so the standard deviation is √100 = 10.

Sources: Expected value — Wikipedia

Frequently asked questions

What does the expected value actually represent?

It is the long-run average outcome, not necessarily a value that can occur on any single trial. A fair die has an expected value of 3.5 even though you can never roll a 3.5 — it is the average you would approach over many rolls.

Why must the probabilities sum to 1?

A valid discrete probability distribution assigns probabilities that cover every possible outcome exactly once, so they must total 1 (100%). If they do not, the outcomes are incomplete or mis-measured; this calculator flags a sum that differs from 1 by more than a small rounding tolerance.

How is expected value used in decisions?

It is the foundation of decision-making under uncertainty — comparing gambles, insurance, or investments by their probability-weighted payoff. But it ignores risk appetite: two options with the same expected value can have very different spreads, which is why the standard deviation matters alongside it.

Related calculators

Probability Calculator

Work out a probability with the steps shown: favorable ÷ total outcomes with odds, two events (and, or, neither, conditional), repeated trials with the binomial distribution, and the area under a normal curve.

Binomial Distribution Calculator

Find binomial probabilities — exactly k, at most k and at least k successes in n independent trials — plus the mean and standard deviation.

Poisson Distribution Calculator

Compute Poisson probabilities — exactly k, at most k and at least k occurrences for a given average rate λ — with the mean and standard deviation.

Standard Deviation Calculator

Paste a list of numbers and get the sample or population standard deviation, variance, mean, standard error, confidence interval, quartiles, z-scores, a frequency table and a histogram, with every step worked out.

Variance Calculator

Paste a list of numbers to get the sample and population variance, plus the standard deviation, mean and count — with the n vs n−1 difference explained.

For education — check assumptions (sample size, distribution) before using statistics in real decisions.